In Judaism what is believed to happen to someone after they die? Is there some idea of an afterlife, or is that purely a Christian invention?
Judaism has from the Torah itself always spoken of a life following this one. The Torah speaks about what seems to be a physical place, called Sheol, to which one "goes down" following this life. A variety of different passages indicate that Sheol was probably though of as located in the center of the earth, although it is no ever formally described. What is clear is that this was a well-known concept amongst the ancient Israelites.
It was not until the Pharisees (c. 100 B.C.E.) that the notion of a spiritual life after death developed in any meaningful way in Jewish thought. The Pharisees, who were the forerunners of the rabbis, taught that when the Torah spoke of reward for following God's ways, the reward would be forthcoming in an afterlife, Olam Ha-Ba (world to come), as they called it.
They further taught that there would be an end of time as we know it, ushered in by the Messiah, and at that time, bodily resurrection would occur (Hebrew, T'chiyat Ha-Metim). While this teaching was an innovation, they insisted that it was rooted in Torah, and quoted extensive proof-texts to make their case. The Pharisees never saw themselves as creating anything new, but unfolding and uncovering that which was already existent but not yet manifest. And while this teaching was of enormous importance, they approached the development of Jewish life as we know it today with this teaching as an ever present backdrop, but not as a primary concern. They consciously chose to de-emphasize the importance of resurrection, favoring the observance of Mitzvot for their own sake. The classic text regarding this matter is paraphrased as follows: "Do not be as ones who labor for their master mindful of the reward that will be coming, but rather as those who serve their master with love and with joy" - the fuller understanding being that if we keep our eyes on the reward, and not the task, we'll never earn the reward. To that end, while there is a good deal in our tradition that refers to life after our deaths in this realm, it is not systematized, and therefore open to much interpretation.
Reform Judaism, while not taking any "official" position on the matter, has for the most part ignored the question, and tended towards the belief that there is no such thing. The attitude of Judaism might best be summed up as "We really do not know, but if there is a life after this one, and a reward for what we do, then surely it will be dependent upon the kind of life we have lived - therefore, let us strive to follow God's path for us as closely and as enthusiastically as possible, for then we will surely know all manner of rewards, especially the one of seeing a world that is a better place for our efforts".